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Chapter 5: Problem 32

Use a double-angle formula to rewrite the expression. $$\cos ^{2} x-\frac{1}{2}$$

Short Answer

Expert verified
The expression \(\cos^2 x - \frac{1}{2}\) can be rewritten as \(\cos(2x)\) using the double-angle formula for cosine.

Step by step solution

01

Recognize the formula

The expression \(\cos^2 x - \frac{1}{2}\) is quite similar to the double-angle formula for cosine, \(\cos(2x) = 2\cos^2(x) - 1\). Therefore, our goal is to write the given expression in the exact form as the right side of the double-angle formula.
02

Solve for the expression

The expression can be written in the required form by multiplying 2 to the each term, which gives \(2\cos^2 x - 1\). This exactly matches the right side of the double-angle formula.
03

Rewrite the expression using the double-angle formula

Replace the expression \(2\cos^2 x - 1\) with \(\cos(2x)\), the left side of the double-angle formula. The expression can now be written as \(\cos(2x)\).

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